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Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2b Structured version   Visualization version   GIF version
Theorem carden2b 9965
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9964 are meant to replace carden 10549 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Proof of Theorem carden2b
StepHypRef Expression
1 cardne 9963 . . . . 5 ((cardβ€˜π΅) ∈ (cardβ€˜π΄) β†’ Β¬ (cardβ€˜π΅) β‰ˆ 𝐴)
2 ennum 9945 . . . . . . . 8 (𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ dom card ↔ 𝐡 ∈ dom card))
32biimpa 476 . . . . . . 7 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 ∈ dom card)
4 cardid2 9951 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
53, 4syl 17 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
6 ensym 9002 . . . . . . 7 (𝐴 β‰ˆ 𝐡 β†’ 𝐡 β‰ˆ 𝐴)
76adantr 480 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 β‰ˆ 𝐴)
8 entr 9005 . . . . . 6 (((cardβ€˜π΅) β‰ˆ 𝐡 ∧ 𝐡 β‰ˆ 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
95, 7, 8syl2anc 583 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
11 cardon 9942 . . . . 5 (cardβ€˜π΄) ∈ On
12 cardon 9942 . . . . 5 (cardβ€˜π΅) ∈ On
13 ontri1 6399 . . . . 5 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄)))
1411, 12, 13mp2an 689 . . . 4 ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
1510, 14sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
16 cardne 9963 . . . . 5 ((cardβ€˜π΄) ∈ (cardβ€˜π΅) β†’ Β¬ (cardβ€˜π΄) β‰ˆ 𝐡)
17 cardid2 9951 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
18 id 22 . . . . . 6 (𝐴 β‰ˆ 𝐡 β†’ 𝐴 β‰ˆ 𝐡)
19 entr 9005 . . . . . 6 (((cardβ€˜π΄) β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ 𝐡) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2017, 18, 19syl2anr 596 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2116, 20nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
22 ontri1 6399 . . . . 5 (((cardβ€˜π΅) ∈ On ∧ (cardβ€˜π΄) ∈ On) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅)))
2312, 11, 22mp2an 689 . . . 4 ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
2421, 23sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) βŠ† (cardβ€˜π΄))
2515, 24eqssd 4000 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
26 ndmfv 6927 . . . 4 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
2726adantl 481 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = βˆ…)
282notbid 317 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (Β¬ 𝐴 ∈ dom card ↔ Β¬ 𝐡 ∈ dom card))
2928biimpa 476 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ Β¬ 𝐡 ∈ dom card)
30 ndmfv 6927 . . . 4 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
3129, 30syl 17 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) = βˆ…)
3227, 31eqtr4d 2774 . 2 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
3325, 32pm2.61dan 810 1 (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  dom cdm 5677  Oncon0 6365  β€˜cfv 6544   β‰ˆ cen 8939  cardccrd 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8706  df-en 8943  df-card 9937
This theorem is referenced by:  card1  9966  carddom2  9975  cardennn  9981  cardsucinf  9982  pm54.43lem  9998  nnadju  10195  nnadjuALT  10196  ficardun  10198  ficardunOLD  10199  ackbij1lem5  10222  ackbij1lem8  10225  ackbij1lem9  10226  ackbij2lem2  10238  carden  10549  r1tskina  10780  cardfz  13940
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